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PERIODIC MOTION                           325
250.  Effect of Damping on the Period.  Substituting the values of a and 6 in the expression for the period,
where P0 is the period for the undamped motion. It is evident from equation (XXVI) that the damping increases the period.
251. Lagrange's Method. In the various pendulum problems which we have discussed the vibrating body was considered to be either a particle or a rigid body. These simplifications were necessary because the methods we have used cannot be applied conveniently to complicated systems. La-grange (1736-1813) introduced into Dynamics a method which can be applied to any vibrating system. The following is a special case of his method adapted to conservative systems which have only one degree of freedom of motion.
Express the potential energy of the system as a function of a properly chosen* coordinate qy so that when expanded in ascending powers of q the first power of q does not appear. Then the potential energy takes the form
U= ft + &22 + ft?3 +    ,         (XXVII)
where ft, ft, etc., are constants.    The constant ft can be
* It is shown in books on advanced Dynamics that such a choice is always possible.